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Pell's equation — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Pell%27s_equation
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T16:30:23.514Z
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Pell's equation — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Pell%27s_equation (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_cd291b52a1ca (Metraipolis archive dc_cd291b52a1ca, sha256 af44eb538fe4a33f102346bc40f18970ebae9074b81598b5405b2f3cad6cd633)

BibTeX

@misc{dccd291b52a1ca,
  title = {Pell's equation — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Pell%27s_equation},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_cd291b52a1ca; sha256 af44eb538fe4a33f102346bc40f18970ebae9074b81598b5405b2f3cad6cd633},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Pell%27s_equation. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x 2 − n y 2 = 1 , {\displaystyle x^{2}-ny^{2}=1,} where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. This kind of equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to 92 x 2 + 1 = y 2 {\displaystyle 92x^{2}+1=y^{2}} in his Brāhmasphuṭasiddhānta circa 628. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Bhaskara II is generally credited with developing the chakravala method, building on the work of Jayadeva and Brahmagupta. Solutions to specific examples of Pell's equation, such as the Pell numbers arising from the equation with n = 2, had been known for much longer, since the time of Pythagoras in Greece and a similar date in India. William Brouncker was the first European to solve Pell's equation. The name of Pell's equation arose from Leonhard Euler mistakenly attributing Brouncker's solution of the equation to John Pell.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «number theory»).
Fonte: Wikipedia in inglese, voce «Pell's equation» — https://en.wikipedia.org/wiki/Pell%27s_equation
Wikidata Q853067 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q853067
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Pell%27s_equation?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1372946865, consultata il 2026-10-07T15:59:02.164Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 24c98299688c44e8675fedf3209996b1ba92fe5a6fed504002640a75f6e1b003